#6589135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,24,31,41)(20,32,35,25)(21,26,30,40)(23,28,37,45)(29,43,34,39)(33,48,47,42)(36,46,38,44);(1,3,5,4)(2,6,8,7)(17,19,22,31)(18,24,27,41)(20,33,35,47)(21,37,30,23)(25,42,32,48)(26,45,40,28)(29,38,34,36)(39,46,43,44);(10,13)(14,16)(17,20,22,35)(18,25,27,32)(19,29,31,34)(21,33,30,47)(23,38,37,36)(24,39,41,43)(26,42,40,48)(28,46,45,44);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,26,27,40)(19,23,31,37)(20,34,35,29)(24,28,41,45)(25,43,32,39)(33,38,47,36)(42,46,48,44);(1,4,5,3)(2,6,8,7)(19,30)(21,31)(23,37)(24,40)(26,41)(28,45)(29,47)(33,34)(36,38)(39,48)(42,43)(44,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496331);